Vertical vibration of a rigid circular body and harmonic rocking of a rigid rectangular body on an elastic stratum

Abstract

The mixed boundary-value problems of the vertical vibration of a rigid circular body and the rocking of a long rigid rectangular body on an infinitely wide elastic stratum have been precisely formulated in terms of dual integral equations. Approximate solutions of these equations for the case of a frictionless foundation base have been obtained by establishing in a novel manner an equivalent system on a semi-infinite elastic medium. It is shown that the response of a body vibrating at frequency factor η2 on a stratum of finite depth is approximately equivalent to that of the body with its inertia increased by a factor η22/η2e2, but vibrating at a lower frequency factor η2=(η22-1/h̃ 2)½ on a semi-infinite medium of the same elastic constants as the stratum of non-dimensional depth h. All the results approach corresponding semi-infinite medium results as the stratum depth tends to infinity. This, therefore, corrects the error of Warburton [1] in which the response of a body on a semi-infinite medium lies between responses on strata of finite depths contrary to the expected asymptotic approach confirmed by the experiments of Arnold et al. [2].Finally, two important results are established for this system: a stratum depth of about five times the base radius (or semi-width, for the rectangular body) is a very fair approximation to a semi-infinite medium; resonant frequency of a body on a stratum decreases with increasing stratum depth. Furthermore, the resonant frequency factor, η2, of bodies with large inertia ratios (greater than about 10) can be estimated from the semi-infinite medium solution irrespective of the stratum depth. The present theory consistently shows good agreement with the experimental results of Arnold et al. [2].